I'm going to leave much of this answer to those responding.
My point here is to show the differing opinions out there and, certainly, share my own.
Whoa, baby, do I have my own...
I'm an American, born-and-bred, growing up on Beatles, cantorial, Walt Disney, zemiros, Dean Martin, Shlomo Carlebach, Peter, Paul and Mary, CSNY, EW&F, ELO, Marvin Gaye, Gerrer and Modzitzer, Sherwood Goffin, Ruach and Joni Mitchell... an often-depressing combination of cultures for a growing musician...and that's just spanning one decade of almost five.
I made a career combining (!) many of these elements. Finding a balance is still an artistic and personal struggle, but it also involves dealing with the constantly-changing morays of Jewish audiences.
My next question will be "what is American Jewish music?" Answering that question (and arguing it, I expect) may very well involve responses to the first one.